Abstract
We investigate the problem of scheduling N jobs on parallel identical machines in J successive stages with finite buffer capacities between consecutive stages in a real-time environment. The objective is to find a schedule that minimizes the sum of weighted completion time of jobs. This problem has proven strongly NP-hard. In this paper, the scheduling problem is formulated as an integer programming model considering buffers as machines with zero processing time. Lagrangian relaxation algorithms are developed combined with a speed-up dynamic programming approach. The complication and time consumption of solving all the subproblems at each iteration in subgradient optimization motivate the development of the surrogate subgradient method, where only one subproblem is minimized at each iteration and an adaptive multiplier update scheme of Lagrangian multipliers is designed. Computational experiments with up to 100 jobs show that the designed surrogate subgradient algorithm provides a better performance as compared to the subgradient algorithm.
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Acknowledgements
This research is partly supported by the National Natural Science Foundation for Distinguished Young Scholars of China (Grant no. 70425003), National Natural Science Foundation of China (Grant no. 60274049) and (Grant no. 70171030), Fok Ying Tung Education Foundation and the Excellent Young Faculty Program of the Ministry of Education, China. We thank the Production Manufacturing Centre in Baosteel for providing a lot of production information and data. We also thank the two anonymous referees for their valuable suggestions. We are also grateful to Chen HX and Zhao X of the University of Connecticut for insightful discussions in algorithm development.
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Appendix A
Appendix A
Proof of Theorem 2
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According to the definition of the surrogate dual function, it can be derived that L̃(π, B)⩾L(π). Although L(π) provides a lower bound on the optimal objective L*, it is not enough to assure that L̃(π, B)⩽L *. Therefore, the surrogate dual value is not necessary to be a lower bound of the optimal cost L*.
Since lim n → ∞ s n=0, in terms of (26) there is always an iteration value n(ɛ) such that ∥π h,i−π (h−1),i∥<ɛ, ∀n⩾n(ɛ). Therefore, there exists a small enough positive vector ɛ such that
That is, the corresponding schedules obtained in the consecutive cycles are identical for n⩾n(ɛ). B (h−1),i is the minimum subproblem solution at π (h−1),i. As a result, the surrogate dual cost is close to the dual cost, resulting in the conclusion that the surrogate dual value is a lower bound on the optimal dual cost for large enough n.
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Tang, L., Xuan, H. Lagrangian relaxation algorithms for real-time hybrid flowshop scheduling with finite intermediate buffers. J Oper Res Soc 57, 316–324 (2006). https://doi.org/10.1057/palgrave.jors.2602033
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DOI: https://doi.org/10.1057/palgrave.jors.2602033